Multidimensional Hypercubes Research Articles

Visibility phenomena in hypercubes

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic themes. For example, we prove that almost all self-visible triangles with vertices in the lattice of points with integer coordinates in W=([0,N]∩Z)d are almost equilateral having all sides almost equal to dN/6, and the sine of the typical angle between rays from the visual spectra from the origin of W is, in the limit, equal to 7/4, as d and N/d tend to infinity. We also show that there exists an interesting number theoretic constant Λd,K, which is the limit probability of the chance that a K-polytope with vertices in the lattice W has all vertices visible from each other.

The development and implementation of a multidimensional relational database of fire danger metrological data

The study designs and implements a multi-dimensional relational database of fire danger metrological information, as well as areas with vegetation. During conceptualization, OLAP algorithms are taken into account as applied to the selection of long-term meteodata and their mathematical processing presented by multidimensional hypercubes. When designing the SQL structure, a layered architecture of geographic information system (GIS) is used, including attribute and geographical characters. The data according to the technology is tested in the Jewish Autonomous Region. The database contains metrological information from five weather stations from 1960 to 2020 and five branches of local forest districts from 1997 to 2020. At present, data is being filled in for the Omsk Region, Primorsky Krai of the Russian Federation and the Osh Region of the Kyrgyz Republic.

Computational Complexity Analysis of Decomposition Methods of OLAP Hyper-cubes of Multidimensional Data

The paper investigates the problems of reduction (decomposition) of multidimensional data models in the form of hypercube OLAP structures. OLAP data processing does not allow changes in the dimension of space. With the increase in data volumes, the productivity of computing cubic structures decreases. Methods for reducing large data cubes to sub-cubes with smaller volumes can solve the problem of reducing computing performance.The reduction problems are considered for cases when the cube lattice has already determined criteria aggregation, and the cube decomposition into smaller cubes is needed to reduce the computation time of the full lattice when dynamically changing data in the cube.The objective of the paper is to find conditions for reducing the computational complexity of solving data analysis problems by reduction methods, to obtain exact quantitative boundaries for reducing the complexity of decomposition methods from the class of polynomial degrees of complexity, to establish the nature of the dependence of computational performance on the structural properties of a hypercube, and to determine the quantitative boundaries of computational performance for solving decomposition problems of data aggregation .The study of the computational complexity of decomposition methods for the analysis of multidimensional hyper-cubes of polynomial-logarithmic and polynomial degrees of complexity is carried out. An exact upper limit is found for reducing the complexity of decomposition methods for analyzing the initial OLAP - data hypercube with respect to non-decomposition ones and based on them criteria are proved for the effective application of reduction methods for analyzing hypercube structures in comparison with traditional non-reduction methods.Examples of decomposition methods of cube structures are presented, both reducing and increasing computational complexity in comparison with calculations using the full model.The results obtained can be used in processing and analysis of information arrays of hypercube structures of analytical OLAP-systems belonging to the BigData class, or ultra-large computer multidimensional data systems.

Construction of Multivariate Interpolation Hermite Polynomials for Finite Element Method

A new algorithm for constructing multivariate interpolation Hermite polynomials in analytical form in a multidimensional hypercube is presented. These polynomials are determined from a specially constructed set of values of the polynomials themselves and their partial derivatives with continuous derivatives up to a given order on the boundaries of the finite elements. The effciency of the finite element schemes, algor thms and programs is demonstrated by solving the Helmholtz problem for a cube.

A generalization of Bernstein-Durrmeyer operators on hypercubes by means of an arbitrary measure

In this paper we introduce and study a sequence of Bernstein-Durrmeyer type operators $(M_{n,\mu})_{n\geq 1}$, acting on spaces of continuous or integrable functions on the multi-dimensional hypercube $Q_d$ of $\mathbf{R}^d$ ($d\geq 1$), defined by means of an arbitrary measure $\mu$. We investigate their approximation properties both in the space of all continuous functions and in $L^p$-spaces with respect to $\mu$, also furnishing some esitmates of the rate of convergence. Further, we prove an asymptotic formula for the $M_{n,\mu}$'s. The paper ends with a concrete example.

ПОСТРОЕНИЕ ЗАЩИЩЕННОЙ СИСТЕМЫ УПРАВЛЕНИЯ МНОГОМЕРНЫМИ СТРУКТУРАМИ ДАННЫХ

Nowadays one should not underestimate importance of informational technologies across all fields of humans’ life. Computational technologies and tools became crucial parts of any human activity. However, since information is becoming a strategic resource, it is necessary to ensure its security by all means. A large number of contemporary studies have been carried on in the field of informational security. Recently, multidimensional information technologies have become popular for data analysis and processing. This has led to a contradiction between ever growing demand of multidimensional data processing systems and inability to properly secure such system due to insufficient research within this field. As a result, sensitive multidimensional data cannot be securely processed due to lack of proper secure systems able to ensure appropriate level of informational security. Authors decided to develop such system. The paper proposes architecture of multidimensional data management system capable of secure information management built up as a multidimensional hypercube. Proposed architecture consists of a description of main system components, a description of task performed by those components and principles of their communication. The paper describes a way to build a system which ensures safe read/write operations for confidential information organized as multidimensional hypercube. The proposed solution allows secure processing of sensitive multidimensional data and therefore extending the area where such systems can be used.

Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators

Deepening the study of a new approximation sequence of positive linear operators we introduced and studied in [12], in this paper we disclose its relationship with the Markov semigroup (pre)generation problem for a class of degenerate second-order elliptic differential operators which naturally arise through an asymptotic formula, as well as with the approximation of the relevant Markov semigroups in terms of the approximating operators themselves.The analysis is carried out in the context of the space C(K) of all continuous functions defined on an arbitrary convex compact subset K of Rd, d≥1, having non-empty interior and a not necessarily smooth boundary, as well as, in some particular cases, in Lp(K) spaces, 1≤p<+∞. The approximation formula also allows to infer some preservation properties of the semigroup such as the preservation of the Lipschitz-continuity as well as of the convexity. We finally apply the main results to some noteworthy particular settings such as balls and ellipsoids, the unit interval and multidimensional hypercubes and simplices. In these settings the relevant differential operators fall into the class of Fleming–Viot operators.

Handling measurement function changes with Slowly Changing Measures

Data Warehouses (DWs) are historical databases on business events, organized as multi-dimensional hypercubes that support analytical decision making. Extract, Transform, and Load (ETL) processes apply measurement functions to compute parameterized scores from the business events, such as sales figures, customer reliability scores, churn likelihood, or sentiment indices. These scores, saved as measures in the DW, serve as basis for analytical reports and corporate performance analysis. Dimensions model subject-oriented data used as analysis perspectives when interpreting the measures. While measures and measurement functions are traditionally regarded as stable within the DW schema, its dimension values commonly change over time.In reality, measures are also subject to change if DW designers (i) tune a parameter of the underlying measurement function, or (ii) update the scoring algorithm as a whole. In both scenarios, the changes must be obvious to the business analysts. Otherwise the changed measure semantics leads to incomparable measure values, and thus unsound and worthless analysis results.To handle measure evolution properly, this paper proposes Slowly Changing Measures (SCMs) as an additional DW modeling concept. Its core idea is a valid time for measurement functions, analogous to dimensions. The paper proposes four increasingly rich SCM types, each specifying a set of options for the change history management of measure definitions. Most of the SCM types provide for probable changes of measurement functions at design time, reducing manual interference to the necessary minimum upon the actual change.Each SCM type is explained in detail, and illustrated using a practical scenario. Furthermore, the paper presents a proof-of-concept prototype based on the TPC-H business model, and implemented in a simple relational database system. The pros and cons of every SCM type are discussed, and recommendations given for their implementation in a practical DW system.

Design and performance of BICM-ID systems with hypercube constellations

This paper introduces new mappings of QPSK symbols, viewed as a multi-dimensional hypercube, to improve the performance of bit-interleaved coded modulation with iterative decoding (BICM-ID). By evaluating the upper bound of the bit error rate performance of BICM-ID, a condition to find the best mapping of a hypercube constellation in terms of the asymptotic performance under different channel models is established. A general and simple algorithm to construct the best mapping of a hypercube is then proposed. Analytical and simulation results show that the use of the proposed mappings together with very simple convolutional codes can offer significant coding gains over the conventional BICM-ID systems for all the channel models considered. Such coding gains are achieved without bandwidth or power expansion and with a very small increase in the system complexity.

Optimized management of large-scale data sets stored on tertiary storage systems

We focus on extending RasDaMan's multidimensional query language using a new concept called object framing. With this extension, users no longer are restricted to performing range queries in the shape of multidimensional hypercubes. It can be able to formulate range queries by indicating complex frames. Thus, object framing represents a generalization of geometric operations. A new system, optimized toward high-performance computing, extends the RasDaMan (Raster Data Management) database management system to allow flexible management of multidimensional spatiotemporal data and to reduce tertiary storage access time.

Concurrent engineering design using interval methods

Modern design practice has displaced serial approaches in favour of concurrent engineering design (CED), which takes several fields simultaneously into consideration, using interaction and iteration to proceed towards generally acceptable solutions. Concurrent design is very often based upon a mathematical model, which not only defines the properties and capabilities of a product, but also economic and production aspects, such as costs, tooling, machine loading and scheduling, and leads to some ‘optimum’ solution satisfying all requirements as closely as possible. This scenario does not reflect actual design activity. Each team member usually has a set of approximate requirements, which can be expressed as a range of admissible values of the design parameters. The aim is to resolve these demands, and determine a set of consistent parameter ranges which then define the design space. Iteration on the implied design alternatives leads to a final solution which is both ‘optimal’ and acceptable. A methodology which is suitable for this purpose is a mathematical model based on interval arithmetic or interval analysis. The intervals for a set of design parameters form a vector which physically defines a multi-dimensional hypercube; the design problem is then studied by a procedures termed hypercube analysis.

Algorithms for multi-extremal mathematical programming problems employing the set of joint space-filling curves

Some powerful algorithms for multi-extremal non-convex-constrained optimization problems are based on reducing these multi-dimensional problems to those of one dimension by applying Peano-type space-filling curves mapping a unit interval on the real axis onto a multi-dimensional hypercube. Here is presented and substantiated a new scheme simultaneously employing several joint Peano-type scannings which conducts the property of nearness of points in many dimensions to a property of nearness of pre-images of these points in one dimension significantly better than in the case of a scheme with a single space-filling curve. Sufficient conditions of global convergence for the new scheme are investigated.

多次元超立方体内の格子点上でのみデータが与えられている場合の最小2乗近似法

多次元超立方体内の格子点上でのみデータが与えられている場合の最小2乗近似法